Lemma 37.11.10. Let $f : X \to Y$ be a morphism of schemes. The following are equivalent:

1. $f$ is formally smooth,

2. for every $x \in X$ there exist opens $x \in U \subset X$ and $f(x) \in V \subset Y$ with $f(U) \subset V$ such that $f|_ U : U \to V$ is formally smooth,

3. for every pair of affine opens $U \subset X$ and $V \subset Y$ with $f(U) \subset V$ the ring map $\mathcal{O}_ Y(V) \to \mathcal{O}_ X(U)$ is formally smooth, and

4. there exists an affine open covering $Y = \bigcup V_ j$ and for each $j$ an affine open covering $f^{-1}(V_ j) = \bigcup U_{ji}$ such that $\mathcal{O}_ Y(V) \to \mathcal{O}_ X(U)$ is a formally smooth ring map for all $j$ and $i$.

Proof. The implications (1) $\Rightarrow$ (2), (1) $\Rightarrow$ (3), and (2) $\Rightarrow$ (4) follow from Lemma 37.11.5. The implication (3) $\Rightarrow$ (4) is immediate.

Assume (4). The proof that $f$ is formally smooth is the same as the second part of the proof of Lemma 37.11.7. Consider a solid commutative diagram

$\xymatrix{ X \ar[d]_ f & T \ar[d]^ i \ar[l]^ a \\ Y & T' \ar[l] \ar@{-->}[lu] }$

as in Definition 37.11.1. We will show the dotted arrow exists thereby proving that $f$ is formally smooth. Let $\mathcal{F}$ be the sheaf of sets on $T'$ of Lemma 37.9.4 as in the special case discussed in Remark 37.9.6. Let

$\mathcal{H} = \mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ T}(a^*\Omega _{X/Y}, \mathcal{C}_{T/T'})$

be the sheaf of $\mathcal{O}_ T$-modules on $T$ with action $\mathcal{H} \times \mathcal{F} \to \mathcal{F}$ as in Lemma 37.9.5. The action $\mathcal{H} \times \mathcal{F} \to \mathcal{F}$ turns $\mathcal{F}$ into a pseudo $\mathcal{H}$-torsor, see Cohomology, Definition 20.4.1. Our goal is to show that $\mathcal{F}$ is a trivial $\mathcal{H}$-torsor. There are two steps: (I) To show that $\mathcal{F}$ is a torsor we have to show that $\mathcal{F}$ locally has a section. (II) To show that $\mathcal{F}$ is the trivial torsor it suffices to show that $H^1(T, \mathcal{H}) = 0$, see Cohomology, Lemma 20.4.3.

First we prove (I). To see this, for every $t \in T$ we can choose an affine open $W \subset T$ neighbourhood of $t$ such that $a(W)$ is contained in $U_{ji}$ for some $i, j$. Let $W' \subset T'$ be the corresponding open subscheme. By assumption (4) we can lift $a|_ W : W \to U_{ji}$ to a $V_ j$-morphism $a' : W' \to U_{ji}$ showing that $\mathcal{F}(W')$ is nonempty.

Finally we prove (II). By Lemma 37.11.8 we see that $\Omega _{U_{ji}/V_ j}$ locally projective. Hence $\Omega _{X/Y}$ is locally projective, see Properties, Lemma 28.21.2. Hence $a^*\Omega _{X/Y}$ is locally projective, see Properties, Lemma 28.21.3. Hence

$H^1(T, \mathcal{H}) = H^1(T, \mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ T}(a^*\Omega _{X/Y}, \mathcal{C}_{T/T'}) = 0$

by Lemma 37.11.9 as desired. $\square$

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